1998/01/01 by Michael J. Mossinghoff · 2 citations
Mathematics · Computer Science · #Meromorphic and Entire Functions #Matrix Theory and Algorithms #semigroups and automata theory #Measure (data warehouse) #Annotation #Algorithm #Mathematics #Degree (music) #Type (biology) #Computer science #Discrete mathematics #Artificial intelligence #Database #Physics
paper · pdf · doi:10.1090/s0025-5718-98-01006-0
openalex publication_date 1998/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
We describe several searches for polynomials with integer coefficients and small Mahler measure. We describe the algorithm used to test Mahler measures. We determine all polynomials with degree at most 24 and Mahler measure less than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1.3"> <mml:semantics> <mml:mn>1.3</mml:mn> <mml:annotation encoding="application/x-tex">1.3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40, and check certain sparse polynomials with height 1 and degree as large as 181. We find a new limit point of Mahler measures near <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1.309"> <mml:semantics> <mml:mn>1.309</mml:mn> <mml:annotation encoding="application/x-tex">1.309</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , four new Salem numbers less than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1.3"> <mml:semantics> <mml:mn>1.3</mml:mn> <mml:annotation encoding="application/x-tex">1.3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and many new polynomials with small Mahler measure. None has measure smaller than that of Lehmer’s degree 10 polynomial.